# 007B Sample Midterm 3, Problem 1 Detailed Solution

Divide the interval  ${\displaystyle [0,\pi ]}$  into four subintervals of equal length   ${\displaystyle {\frac {\pi }{4}}}$   and compute the right-endpoint Riemann sum of  ${\displaystyle y=\sin(x).}$

Background Information:

1. The height of each rectangle in the right-hand Riemann sum

is given by choosing the right endpoint of the interval.

Solution:

Step 1:
Let  ${\displaystyle f(x)=\sin(x).}$
Each interval has length  ${\displaystyle {\frac {\pi }{4}}.}$
Therefore, the right-endpoint Riemann sum of  ${\displaystyle f(x)}$  on the interval  ${\displaystyle [0,\pi ]}$  is

${\displaystyle {\frac {\pi }{4}}{\bigg (}f{\bigg (}{\frac {\pi }{4}}{\bigg )}+f{\bigg (}{\frac {\pi }{2}}{\bigg )}+f{\bigg (}{\frac {3\pi }{4}}{\bigg )}+f(\pi ){\bigg )}.}$

Step 2:
Thus, the right-endpoint Riemann sum is

${\displaystyle {\begin{array}{rcl}\displaystyle {{\frac {\pi }{4}}{\bigg (}\sin {\bigg (}{\frac {\pi }{4}}{\bigg )}+\sin {\bigg (}{\frac {\pi }{2}}{\bigg )}+\sin {\bigg (}{\frac {3\pi }{4}}{\bigg )}+\sin(\pi ){\bigg )}}&=&\displaystyle {{\frac {\pi }{4}}{\bigg (}{\frac {\sqrt {2}}{2}}+1+{\frac {\sqrt {2}}{2}}+0{\bigg )}}\\&&\\&=&\displaystyle {{\frac {\pi }{4}}({\sqrt {2}}+1).}\\\end{array}}}$

${\displaystyle {\frac {\pi }{4}}({\sqrt {2}}+1)}$